If you haven't already, maybe worth a quick look to see if any of them could be 'modified' to power the fan in standby.
It's hardware. Unless you fancy rerouting the fan power to the standby rail, forget it (plus, of course, the PWM controller will have been shut down, so a bypass for that would be required...).
I'm a little puzzled. Why should the HDD temperature continue to rise after entering standby?
Regardless of previous answers to this, the HDD (and other internal components) act as a reservoir of heat. Once the fan stops and the HDR becomes an essentially sealed unit, the heat will try to even out even out by a diffusion process very similar to flow of electrons through a conductive medium. As the outside of the HDD is no longer being actively cooled, its thermal resistance is now greater and the skin temperature can indeed rise (although the internal temperature should decrease if there is nothing hotter elsewhere to flow to it). You must regard total heat as the quantity, spot temperatures are like voltage in an electrical circuit (voltage differences define the direction of current flow, not how much charge there is).
When the system reaches equilibrium, it will be at room temperature throughout. Until then, the heat has to dissipate by whatever means it can find - conduction, radiation, convection, and forced cooling (but there isn't any of that, and very limited convection). Essentially the heat will find its way to the outer case by conduction and internal radiation (and maybe a little convection, but with little temperature gradient there is not much to drive convection), and from there to the outside world by radiation and convection.
Similar happens in a car engine bay if you stop (killing air flow from motion) and turn off the engine (killing the rad fan, if yours does not run in standby). The heat inside the engine will continue to dissipate, and would be capable of boiling the coolant even after stopping if the coolant wasn't pressurised.